Abstract

Coherent X-ray diffraction imaging (CXDI) of the displacement field and strain distribution of nanostructures in kinematic far-field conditions requires solving a set of non-linear and non-local equations. One approach to solving these equations, which utilizes only the object’s geometry and the intensity distribution in the vicinity of a Bragg peak as a priori knowledge, is the HIO+ER-algorithm. Despite its success for a number of applications, reconstruction in the case of highly strained nanostructures is likely to fail. To overcome the algorithm’s current limitations, we propose the
HIOORM+ERM-algorithm which allows taking advantage of additional a priori knowledge of the local scattering magnitude and remedies HIO+ER’s stagnation by incorporation of randomized overrelaxation at the same time. This approach achieves significant improvements in CXDI data analysis at high strains and greatly reduces sensitivity to the reconstruction’s initial guess. These benefits are demonstrated in a systematic numerical study for a periodic array of strained silicon nanowires. Finally, appropriate treatment of reciprocal space points below noise level is investigated.

Figures (5)

Schematics of the
HIOORM-algorithm according to Eqs. (7a) and (11): In addition to QΓ;λΓ in the HIOOR-algorithm the operator MM is applied before the calculation of the next iterative candidate
ρeff(i+1)(x) is performed. In the limit ML,n → 0 and MH,n → ∞ for all n, the algorithm reduces to HIOOR because MM → 1.

Illustration of the physical system used for our investigation of the
HIOORM+ERM-algorithm. Figure (a) shows the geometry and composition of the upper region of a periodic unit of the periodic Si-nanowire system. The hatched domain will become important in next sections. Figures (b)–(d) depict the phase field QB · u(x) of effective electron density ρeff(x) for this system for different values of the maximum strain εM on the wires’ symmetry axis in the crystalline domain, if the Bragg peak QB = [004] is investigated. Figures (e)–(g) show the scattering signal in presence of the low signal cutoff ΓN (see Sec. 1.4). Only the central region of the scattering signal around the Bragg peak QB is shown (QB is located at the yellow dot). Data points below the noise level ΓN are masked in dark cyan.

Behavior of the
HIOORM+ERM-algorithm (including its limiting cases without randomization and/or without constraints on the local scattering magnitude, see grey boxes with black border at the beginning of each row of four subfigures) for ideal data: Depicted are two-dimensional cuts of the success rate s through the three dimensional parameter space for either fixed strain εM, number of iterations i or angle φMax. We count all random initial trails as success for which the angle φ(i) (defined in Eq. (13)) to the reference solution ρeff is below φMax in iteration (i) for that particular value of strain εM. The domain ΩSub corresponds to the hatched domain in Fig. 2(a).

Characteristics of the
HIOORM+ERM-algorithm in presence of a low cutoff ΓN. Figures (a) and (b) compare the success rate s of the models defined in Sec. 1.4 for fixed strain εM = {0.20%, 0.28%} after i = 500 iterations. Figures (c)–(f) contain two-dimensional plots of the success rate s for model (E) (analogue to Fig. 3). For Figs. (a), (c) and (d), no constraints on ζ have been applied. Figures (b), (e) and (f) illustrate the improvement if bounds on the local scattering magnitude are taken into account. Note the different range of the strain axes.

Summary of our numerical investigation: Starting from standard HIO+ER, the improvements which are achieved by randomized overrelaxation and the additional constraints M on the local scattering magnitude ζ are simplified to two values. The upper value in each rectangular box is the maximum strain εM for which an almost perfect solution could be reconstructed within i = 500 iterations and for almost all random initial trails. The lower value is the maximum strain εM for which the reconstruction of the effective electron density
ρeff(500) was successful if the requirement for success is relaxed: Any strain εM for which at least some non-negligible fraction of initial guesses managed to achieve a result close to the solution ρeff (φMax ≲ 10.0°) is classified as suitable for the respective approach and constraints.